Abstract

We consider second-quantized homogeneous Bose gas in a large cubic box with periodic boundary conditions at zero temperature. We discuss the energy-momentum spectrum of the Bose gas and its physical significance. We review various rigorous and heuristic results as well as open conjectures about its properties. Our main aim is to convince the readers, including those with mainly mathematical background, that this subject has many interesting problems for rigorous research. In particular, we investigate the upper bound on the infimum of the energy for a fixed total momentum given by the expectation value of one-particle excitations over a squeezed states. This bound can be viewed as a rigorous version of the famous Bogoliubov method. We show that this approach seems to lead to a (nonphysical) energy gap. The variational problem involving squeezed states can serve as the preparatory step in a perturbative approach that should be useful in computing excitation spectrum. This version of a perturbative approach to the Bose gas seems (at least in principle) superior to the commonly used approach based on the -number substitution.

Received 18 December 2008Accepted 14 April 2009Published online 16 June 2009

Acknowledgments:

The research of J.D. is supported in part by the Polish Ministry of Science and Higner Education under Grant No. N201 270135 and was partly done while the author was visiting the Institute for Mathematical Sciences, National University of Singapore in 2008. The visit was supported by the Institute. A part of the work of H.C. was done during his visit to the Erwin Schrödinger Institute. H.C. and J.D. also acknowledge support from the Danish F.N.U. grant Mathematical Physics.

The authors acknowledge interesting and useful discussions with J. P. Solovej, J. Yngvason, and V. Zagrebnov. They are also grateful to the referees for their remarks.